tokfandomcom-20200215-history
Sesquilinear form
In , a sesquilinear form is a generalization of a that, in turn, is a generalization of the concept of the of . A bilinear form is in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a manner, thus the name; which originates from the Latin meaning "one and a half". The basic concept of the dot product – producing a from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of what a vector is. A motivating special case is a sesquilinear form on a , . This is a map that is linear in one argument and "twists" the linearity of the other argument by (referred to as being in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any and the twist is provided by a . An application in requires that the scalars come from a (skewfield), , and this means that the "vectors" should be replaced by elements of a -module}}. In a very general setting, sesquilinear forms can be defined over -modules for arbitrary . Convention Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by mathematical physicists and originates in in . In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear. Complex vector spaces Over a a map is sesquilinear if : \begin{align} &\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w)\\ &\varphi(a x, b y) = \overline{a}b\,\varphi(x,y)\end{align} for all and all . }} is the complex conjugate of . A complex sesquilinear form can also be viewed as a complex : \overline{V} \times V \to \mathbf{C} where }} is the to . By the of s these are in one-to-one correspondence with complex linear maps : \overline{V} \otimes V \to \mathbf{C}. For a fixed in the map is a on (i.e. an element of the ). Likewise, the is a on . Given any complex sesquilinear form on we can define a second complex sesquilinear form via the : : \psi(w,z) = \overline{\varphi(z,w)}. In general, and will be different. If they are the same then is said to be Hermitian. If they are negatives of one another, then is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form. Matrix representation If is a finite-dimensional complex vector space, then relative to any }} of , a sesquilinear form is represented by a , by the column vector , and by the column vector : : \varphi(w,z) = \varphi \left(\sum_i w_i e_i, \sum_j z_j e_j \right) = \sum_i \sum_j \overline{w_i} z_j \varphi(e_i, e_j) = {\overline{\mathbf{w}}}^\mathrm{T} \mathbf{\Phi} \mathbf{z} . The components of are given by . Hermitian form :''The term '''Hermitian form' may also refer to a different concept than that explained below: it may refer to a certain on a .'' A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form such that : h(w,z) = \overline{h(z, w)}. The standard Hermitian form on is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by : \langle w,z \rangle = \sum_{i=1}^n \overline{w_i} z_i. More generally, the on any complex is a Hermitian form. A minus sign is introduced in the Hermitian form w w^* - z z^* to define the group . A vector space with a Hermitian form is called a '''Hermitian space'. The matrix representation of a complex Hermitian form is a . A complex Hermitian form applied to a single vector : |z|_h = h(z, z) is always . One can show that a complex sesquilinear form is Hermitian the associated quadratic form is real for all . Skew-Hermitian form A complex '''skew-Hermitian form' (also called an antisymmetric sesquilinear form), is a complex sesquilinear form such that : s(w,z) = -\overline{s(z, w)}. Every complex skew-Hermitian form can be written as }} times a Hermitian form. The matrix representation of a complex skew-Hermitian form is a . A complex skew-Hermitian form applied to a single vector : |z|_s = s(z, z) is always pure . Over a division ring This section applies unchanged when the division ring is commutative. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions. Definition A ' -sesquilinear form over a right -module is a with an associated of a such that, for all and all , : \varphi(x \alpha, y \beta) = \sigma(\alpha) \, \varphi(x, y) \, \beta . The associated anti-automorphism for any nonzero sesquilinear form is uniquely determined by . Orthogonality Given a sesquilinear form over a module and a subspace of , the orthogonal complement of with respect to is : W^{\perp}=\{\mathbf{v} \in M \mid \varphi (\mathbf{v}, \mathbf{w})=0,\ \forall \mathbf{w}\in W\} . Similarly, is '''orthogonal' to with respect to , written (or simply if can be inferred from the context), when . This need not be , i.e. does not imply (but see '' below). Reflexivity A sesquilinear form is reflexive if, for all , : \varphi(x, y) = 0 implies \varphi(y, x) = 0 . That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric. Hermitian variations A -sesquilinear form is called -Hermitian if there exists such that, for all , : \varphi(x, y) = \sigma ( \varphi (y, x)) \, \varepsilon . If , the form is called -''Hermitian'', and if , it is called -''anti-Hermitian''. (When is implied, respectively simply Hermitian or anti-Hermitian.) For a nonzero -Hermitian form, it follows that, for all , : \sigma ( \varepsilon ) = \varepsilon^{-1} : \sigma ( \sigma ( \alpha ) ) = \varepsilon \alpha \varepsilon^{-1} . It also follows that is a of the map . The fixed points of this map from a of the of . A -Hermitian form is reflexive, and every reflexive -sesquilinear form is -Hermitian for some . In the special case that is the (i.e., ), is commutative, is a bilinear form and . Then for the bilinear form is called symmetric, and for is called skew-symmetric. Example Let be the three dimensional vector space over the , where is a . With respect to the standard basis we can write and and define the map by: : \varphi(x, y) = x_1 y_1{}^q + x_2 y_2{}^q + x_3 y_3{}^q. The map is an involutory automorphism of . The map is then a -sesquilinear form. The matrix associated to this form is the . This is a Hermitian form. In projective geometry In a , a of the subspaces that inverts inclusion, i.e. : for all subspaces , of , is called a . A result of Birkhoff and von Neumann (1936) shows that the correlations of projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form is nondegenerate if for all in (if and) only if . To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by a , extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by -module}}s. (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.) Over arbitrary rings The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings. Let be a ring, an -module and an of . A map is ' -sesquilinear''' if : \varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w) : \varphi(c x, d y) = c \, \varphi(x,y) \, \sigma(d) for all and all . An element is orthogonal to another element with respect to the sesquilinear form (written ) if . This relation need not be symmetric, i.e. does not imply . A sesquilinear form is '''reflexive' (or orthosymmetric) if implies for all . A sesquilinear form is '''Hermitian' if there exists such that : \varphi(x, y) = \sigma(\varphi(y, x)) for all . A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism is an involution (i.e. of order 2). Since for an antiautomorphism we have for all in , if , then must be commutative and is a bilinear form. In particular, if, in this case, is a skewfield, then is a field and is a vector space with a bilinear form. An antiautomorphism can also be viewed as an isomorphism of , the ''opposite ring based on the same set with the same addition, but whose multiplication operation ( ) is defined by , where the product on the right is the product in . It follows from this that a right (left) -module can be turned into a left (right) -module, . Thus, the sesquilinear form can be viewed as a bilinear form . References Category:Advanced mathematics